We obtain lower bounds on the
rank of the kappa ring \kring of the Delign-Mumford
compactification of the moduli space of curves in different degrees. For this purpose,
we introduce a quotient κc*(\Mgnbar) of \kring, and show that
the rank of this latter ring in degree d is bounded
below by |\PP(d,3g−2+n−d)| where \PP(d,r) denotes the set of partitions
of the positive integer d into at most r parts. In codimension
1 (i.e. d=3g−4+n) we show that the rank of κc*(\Mgnbar)
is equal to n−1 for g=1, and
is equal to
for g > 1. Furthermore, in codimension e=3g−3+n−d, the rank of
κc*(\Mgnbar) (as g and e remain fixed and n grows large)
is asymptotic to
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